the infimum being taken over all \(\mu \in \mathcal {E}^+_A\)  with \(\mu (\mathbb {R}^n)=1\) . If \(c_\alpha (A)<\infty \) , there exists the unique solution \(\gamma _A\)  to the minimum \(\alpha \) -Riesz energy problem over the class of all \(\nu \in \mathcal {E}^+_\alpha \)  such that \(\kappa _\alpha \nu \geqslant 1\)  n.e. on \(A\)  (see e.g. [1]}). This \(\gamma _A\)  is said to be the inner \(\alpha \) -Riesz equilibrium measure for \(A\) ,We also refer to the author's recent work [2]} providing a number of alternative characterizations of the inner capacity \(c_\alpha (A)\)  and the inner equilibrium measure \(\gamma _A\) , the results in [2]} being actually valid even for quite a large class of general function kernels on locally compact spaces. and it satisfies the relations
\(\gamma _A(\mathbb {R}^n)=\kappa _\alpha (\gamma _A,\gamma _A)=c_\alpha (A)\in [0,\infty ),\\\kappa _\alpha \gamma _A=1\text{ \ n.e.\ on $A$}.\) 
